Small subgroup attack in Mozilla NSS

Type intothesymmetry
Reporter Antonio Sanso (
Modified 2016-10-24T06:29:52


tl;dr While the TLS servers attacks has been pretty much studied and fixed (see e.g. and the situation with the TLS clients is (was) not ideal and can be improved. Here I report a Small subgroup attack for TLS clients that I performed against various browsers and reported.

Whoever reads this blog is used to read about OAuth .
For once (and maybe more in the future) let's hijack the usual topic and let's talk about my new "passion" : TLS in particular Diffie–Hellman (DH from now on).

Now, before to start I need to clarify one thing IANAC (I am not a cryptographer) so I might likely end up writing a bunch of mistakes in this blog post...

Diffie-Hellman is used in SSL/TLS, as "ephemeral Diffie-Hellman" (EDH) and it is probably going to be kill soonish (or at least is the intent of Google Chrome). FWIW I personally agree with this unless EDH implements the Negotiated Finite Field specification.

Now in the last years there were at least a couple of issue that affected EDH:

But here the details:

When using TLS_DHE_RSA_WITH_AES_128_CBC_SHA Firefox/Chrome doesn't accept degenerate public key of value 0,1 and -1 since this key lead to pms that is {0,1, -1}.
This (the -1 case) is probably a consequence of CVE-2014-1491 (raised as part of the Triple Handshake Attack ).

I would refer to the classic Diffie Hellman nomenclature

  • p as the prime number
  • g the generator with order p-1 = q
  • y public key
  • x private key


If (p-1)/4 = 0 (mod p) then if I choose my private key x = (p-1)/4 then my public key
y = g^x will generates a prime-order subgroup of size 4.

This means that Mozilla/Chrome will agree on a pms = 1 one time out of 4.

The issue

I set up a server with

p = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084241
g = 3
q =1

and TLS_DHE_RSA_WITH_AES_128_CBC_SHA as cipher.

During the negotiation with Chrome I always choose

x= (p-1)/4 = 3351951982485649274893506249551461531869841455148098344430890360930441007518386744200468574541725856922507964546621512713438470702986642486608412251521060

and pass

y = 11130333445084706427994000041243435077443611277989851635896953056790400956946719341695219235480436483595595868058263313228038179294276393680262837344694991

Chrome/Firefox will happily "agree" on those 4 pms

  • 1
  • 2277474484857890671580024956962411050035754542602541741826608386931363073126827635106655062686466944094435990128222737625715703517670176266170811661389250
  • 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084240
  • 11130333445084706427994000041243435077443611277989851635896953056790400956946719341695219235480436483595595868058263313228038179294276393680262837344694991

Of course the "worse" one is 1 and happens to be 1 time out of 4 (according to Adam Langley though "here's nothing special about sending an odd DH value, it could equally well make its DH private key equal to 42"). So not big deal :(

Just for the record even the easier suggestion given in [1] aka

"Make sure that g^x,g^y and g^xy do not equal to 1"

is not followed and this happens with very high probability (25%)

The Summary